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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.7.75
Chapter 4, Problem 4.7.75

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dy/dx = 3x⁻²ᐟ³, y(−1) = −5

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1
Identify the given differential equation and initial condition: \( \frac{dy}{dx} = 3x^{-\frac{2}{3}} \) with \( y(-1) = -5 \).
Rewrite the differential equation in integral form: \( dy = 3x^{-\frac{2}{3}} dx \).
Integrate both sides with respect to \( x \): \( y = \int 3x^{-\frac{2}{3}} dx + C \), where \( C \) is the constant of integration.
Use the power rule for integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), making sure to add 1 to the exponent \( -\frac{2}{3} + 1 = \frac{1}{3} \).
Apply the initial condition \( y(-1) = -5 \) to solve for \( C \) by substituting \( x = -1 \) and \( y = -5 \) into the integrated expression.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Separable Differential Equations

A separable differential equation can be written as a product of a function of x and a function of y, allowing the variables to be separated on opposite sides of the equation. This technique simplifies solving by integrating each side independently.
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Integration of Power Functions

Integrating power functions involves applying the power rule for integration, which states ∫x^n dx = (x^(n+1))/(n+1) + C, for n ≠ -1. Understanding how to handle fractional and negative exponents is essential for solving the given differential equation.
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Representing Functions as Power Series

Initial Conditions and Particular Solutions

Initial conditions specify the value of the solution at a particular point, allowing determination of the constant of integration. This transforms the general solution of a differential equation into a unique particular solution that satisfies the given condition.
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Related Practice
Textbook Question

101. In Exercises 101 and 102, the graph of f' is given. Determine x-values corresponding to local minima, local maxima, and inflection points for the graph of f.

Textbook Question

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

7. y=sin|x|, -2π≤x≤2π

Textbook Question

Find values of a and b such that the function


ƒ(𝓍) = (a𝓍 + b) / 𝓍² ―1)


has a local extreme value of 1 at 𝓍 = 3. Is this extreme value a local maximum or a local minimum? Give reasons for your answer.

Textbook Question

Theory and Examples


[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.


h(x) = |x + 2| − |x − 3|, −∞ < x < ∞

Textbook Question

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

4. y=9/14x^(1/3)(x^2-7)

Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(1 + tan²θ)dθ (Hint:1 + tan²θ = sec²θ)